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Question 7:

The term "Freshman 15" refers to the claim that college students typically gain 15 pounds during their freshman year. Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 2.3 pounds and a standard deviation of 10.9 pounds.

1. Find the probability that a randomly selected male college student gains 15 pounds or more during their freshman year. What does the result suggest about the validity of the "Freshman 15" claim?

Round your answer to one decimal place as needed.

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Question 10:

A boat was rated to carry 50 passengers based on an assumption of a mean passenger weight of 137 pounds (with the load limit set at 6,650 pounds). After an incident where the boat sank, the assumed mean weight for similar boats was changed from 137 pounds to 174 pounds.

a. The boat is loaded with 50 passengers, and it is assumed that the weights of people are normally distributed with a mean of 176.9 pounds and a standard deviation of 40.8 pounds. Find the probability that the boat is overloaded because the passengers have a mean weight greater than 137 pounds.

Round your answer to four decimal places as needed.

b. The town decided to convert to a maximum capacity of 17 passengers, and the ordinance changed to a load limit of 2,007 pounds. Find the probability that the boat exceeds its maximum capacity due to the mean weight of the passengers being greater than 171 pounds.

Round your answer to four decimal places as needed.

Answer :

Final Answer:

1. Probability of gaining 15+ pounds during freshman year: 0.3446.

2. Probability of boat overload due to passenger weights exceeding 137 pounds: 0.8962.

Explanation:

1. To find the probability of a male college student gaining 15 or more pounds during their freshman year, we use the z-score formula:

Z = (X - μ) / σ

Plugging in the values: X (15 pounds), μ (2.3 pounds), and σ (10.9 pounds), we calculate the z-score:

Z = (15 - 2.3) / 10.9 ≈ 1.2202

Using a standard normal distribution table or calculator, we find the corresponding probability to be approximately 0.3446. This suggests that about 34.46% of male college students gain 15 pounds or more during their freshman year.

2. For the boat overload probability, we calculate the z-score using the new mean weight of 176.9 pounds, standard deviation of 408 pounds, and assumed capacity of 137 pounds:

Z = (137 - 176.9) / 408 ≈ -0.0978

The corresponding probability from the standard normal distribution is approximately 0.4630. To find the probability that the average weight exceeds 137 pounds, we subtract this value from 1:

Probability = 1 - 0.4630 ≈ 0.8962

This indicates that the probability of the boat being overloaded is around 89.62%.

The calculated probabilities provide insights into the likelihood of weight gain among college students and the potential overload of the boat based on the given parameters and assumptions.

Learn more about Probability

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